Find $\mathrm{Gal}(L, \mathbb{Q})$ and prove that $\sqrt[3]{3} \in L$

Question

I am really struggling on these consecutive questions on Galois Theory:

Question 1: Suppose $L$ is a normal closure of $\mathbb{Q}(\sqrt[3]{2}, \sqrt[3]{5}) $ over $\mathbb{Q}$. Find $\mathrm{Gal}(L/\mathbb{Q})$.

Would this be the cyclic product $\mathbb{Z_2} \times \mathbb{Z_3} $ or $\mathbb{Z_6}$?

Question 2: Apply Galois theory to prove that $\sqrt[3]{3} \not\in L$.

$\sqrt[3]{3}$ clearly does not belong to $\mathbb{Q}$ and cannot be obtained using a composition of typical operations (addition, subtraction, multiplication etc) of $\sqrt[3]{2} $ and $\sqrt[3]{5} $ but how can I show this using Galois theory?

Many thanks for your help :-)

Answer 1

Question 1 :

The possible values by an $L$-automorphism for $\sqrt[3]{2}$ are $\sqrt[3]{2}$, $j\sqrt[3]{2}$ and $j^2\sqrt[3]{2}$. And for $\sqrt[3]{5}$ it is $\sqrt[3]{5}$, $j\sqrt[3]{5}$ and $j^2\sqrt[3]{5}$. But the complex $j$ appears in our expressions, so the closure of $\mathbb{Q}(\sqrt[3]{2}, \sqrt[3]{5})$ is $\mathbb{Q}(\sqrt[3]{2}, \sqrt[3]{5},j)$. And the possible images of $j$ by an $L$-automorphism are $j$ and $j^2$. So by considering all the combinations possible, there is a total of 18 possible $L$-automorphisms, and 17 of these automorphisms are of order 3, 17 are of order 6, one is of order 2 and there is $\mathrm{Id}$, so $\mathrm{Gal}(L/\mathbb{Q})$ is isomorphic with $S_3 \times \mathbb{Z_3}$.